Finite Group Databasebeta

← #13161975

Subgroups — #13161975

Order 1728.

910 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic
2#11723656C212yesmaximal, complemented
3C3×#63660C212yesmaximal, complemented
4#11722074C212yesmaximal, complemented
5#10591504—31nomaximal
6#10591501—32nomaximal
7C2×C33×D4C2211yescharacteristic
8C2×C3×#638C412yes—
9C22×C3×C3⋊Dic3C412yes—
10C3×#3830—22no—
11C3×#3830—22no—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C3×#3748C2214yes—
15#88809—22no—
16#88808—22no—
17C2×#3830D311yescharacteristic
18C2×C3×#237—32no—
19C2×C3×#237—61no—
20#63818—32no—
21D3×#208—34no—
22#63660D312yescomplemented
23C3×#833—32no—
24C3×#833—32no—
25C3×#833—62no—
26#63561—32no—
27#63560—34no—
28C23×C33C2×C412yes—
29C22×C3×C3⋊D3C2×C411yescharacteristic
30C33×D4—22no—
31C2×C33×C4C2311yescharacteristic
32C3×#638—24no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3C2×C412yes—
36C2×C3×C3⋊Dic3C2×C411yescharacteristic
37C3×C4×C3⋊D3D412yes—
38C3×C4×C3⋊D3D412yes—
39C3×#634—22no—
40C3×C3⋊(C3⋊C8)—24no—
41#5237—91no—
42#4745—92no—
43C2×C32×D4—21no—
44C2×C32×D4—22no—
45C2×C32×D4—41no—
46C2×C32×D4C2×D312yes—
47C2×C32×D4C2×D311yescharacteristic
48C2×#638Dic312yes—
49C22×C3⋊Dic3Dic312yes—
50#3830—22no—
51#3830—22no—
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C2×#634C2×D311yescharacteristic
54C2×C3×#67—34no—
55C2×C3×#67—62no—
56C22×C3×Dic3—34no—
57C22×C3×Dic3—62no—
58C3×#237—64no—
59C3×#237—64no—
60C3×#237—64no—
61C3×#237—64no—
62C2×C3×C4×D3—32no—
63C2×C3×C4×D3—61no—
64C2×C3×Dic6—32no—
65C2×C3×Dic6—61no—
66#3748C2×D314yes—
67C3×#208—34no—
68C3×#208—34no—
69C3×#208—64no—
70#3712—62no—
71#3711—64no—
72#3710—62no—
73D3×C3⋊C8—64no—
74C22×C33—22no—
75C22×C33—42no—
76C22×C33C22×C411yescharacteristic
77C2×C3×C3⋊D3—41no—
78C2×C3×C3⋊D3C22⋊C412yes—
79C33×C4C2×D412yes—
80C3×C3⋊Dic3—22no—
81C3×C3⋊Dic3C22⋊C411yescharacteristic
82C3×C3⋊Dic3C22⋊C411yescharacteristic
83C2×C3×#41—91no—
84C2×#237—32no—
85C2×#237—61no—
86C2×#208—92no—
87D3×M16—94no—
88C3×#100—92no—
89#836—92no—
90#833—32no—
91#833—32no—
92#833—62no—
93#824—94no—
94C23×C32—24no—
95C23×C32—22no—
96C23×C32—42no—
97C23×C32C4×D314yes—
98C23×C32C2×Dic312yes—
99C22×C3⋊D3C2×Dic311yescharacteristic
100C22×C3×D3—32no—
101C22×C3×D3—61no—
102C32×D4—22no—
103C32×D4—24no—
104C32×D4—24no—
105C32×D4—44no—
106C32×D4—44no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4C22×D312yes—
111C2×C32×C4C22×D311yescharacteristic
112#638—24no—
113C2×C3⋊Dic3—22no—
114C2×C3⋊Dic3—22no—
115C2×C3⋊Dic3C2×Dic312yes—
116C2×C3⋊Dic3C2×Dic311yescharacteristic
117C4×C3⋊D3#6712yes—
118C4×C3⋊D3#6712yes—
119#634—22no—
120C3×#67—68no—
121C3×#67—68no—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—34no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—64no—
128C2×C3×Dic3—64no—
129C2×C3×Dic3—64no—
130C3×C4×D3—34no—
131C3×C4×D3—34no—
132C3×C4×D3—62no—
133C3×C4×D3—62no—
134C3×Dic6—64no—
135C3×Dic6—64no—
136C3⋊(C3⋊C8)—24no—
137C3×C3⋊C8—64no—
138C3×C3⋊C8—64no—
139C3×C3⋊C8—124no—
140#413—271noSylow
141C2×C33—21no—
142C2×C33—42no—
143C2×C33C2×C22⋊C411yescharacteristic
144C3×C3⋊D3—22no—
145C3×#41—182no—
146C3×#41—182no—
147C2×C3×Q8—91no—
148C2×C3×D4—21no—
149C2×C3×D4—22no—
150C2×C3×D4—41no—
151C2×C3×D4—92no—
152C2×C3×D4D3212yes—
153C2×C3×D4D3211yescharacteristic
154C22×C3×C4—91no—
155C22×C3×C4—92no—
156C2×#67—34no—
157C2×#67—62no—
158C22×Dic3—34no—
159C22×Dic3—62no—
160#237—64no—
161#237—64no—
162#237—64no—
163#237—64no—
164C2×C4×D3—32no—
165C2×C4×D3—61no—
166C2×Dic6—32no—
167C2×Dic6—61no—
168C3×M16—94no—
169#208—34no—
170#208—34no—
171#208—64no—
172#208—182no—
173C2×C3⋊C8—182no—
174ΓL(1,𝔽25)—184no—
175D3×C8—184no—
176C22×C32—22no—
177C22×C32—24no—
178C22×C32—24no—
179C22×C32—22no—
180C22×C32—21no—
181C22×C32—41no—
182C22×C32—44no—
183C22×C32—44no—
184C22×C32—42no—
185C22×C32—44no—
186C22×C32—44no—
187C22×C32—84no—
188C22×C32—84no—
189C22×C32C2×C4×D312yes—
190C22×C32C22×Dic311yescharacteristic
191C2×C3⋊D3—41no—
192C2×C3⋊D3#21712yes—
193C2×C3×D3—34no—
194C2×C3×D3—62no—
195C2×C3×D3—122no—
196C2×C3×D3—122no—
197C32×C4—24no—
198C32×C4—22no—
199C32×C4—42no—
200C32×C4D3×D414yes—
201C32×C4C2×#6712yes—
202C3⋊Dic3—22no—
203C3⋊Dic3#21711yescharacteristic
204C3⋊Dic3#21711yescharacteristic
205C3×Dic3—32no—
206C3×Dic3—32no—
207C3×Dic3—61no—
208C3×Dic3—61no—
209C3×Dic3—64no—
210C3×Dic3—64no—
211C2×#41—91no—
212C2×M16—272no—
213#101—272no—
214#100—92no—
215C33#41311yescharacteristic, complemented, Sylow
216C23×C3—22no—
217C23×C3—24no—
218C23×C3—42no—
219C23×C3—91no—
220C23×C3D3×Dic314yes—
221C23×C3#62412yes—
222C22×D3—32no—
223C22×D3—61no—
224C3×Q8—182no—
225C3×D4—24no—
226C3×D4—24no—
227C3×D4—22no—
228C3×D4—44no—
229C3×D4—44no—
230C3×D4—184no—
231C2×C3×C4—22no—
232C2×C3×C4—21no—
233C2×C3×C4—41no—
234C2×C3×C4—91no—
235C2×C3×C4—92no—
236C2×C3×C4—92no—
237C2×C3×C4—92no—
238C2×C3×C4—182no—
239C2×C3×C4—182no—
240C2×C3×C4C2×D3212yes—
241C2×C3×C4C2×D3211yescharacteristic
242#67—68no—
243#67—68no—
244C2×Dic3—32no—
245C2×Dic3—34no—
246C2×Dic3—62no—
247C2×Dic3—61no—
248C2×Dic3—64no—
249C2×Dic3—64no—
250C2×Dic3—64no—
251C2×Dic3—64no—
252C4×D3—34no—
253C4×D3—34no—
254C4×D3—62no—
255C4×D3—62no—
256Dic6—64no—
257Dic6—64no—
258C3×C8—184no—
259C3⋊C8—64no—
260C3⋊C8—64no—
261C3⋊C8—124no—
262C2×C32—21no—
263C2×C32—22no—
264C2×C32—22no—
265C2×C32—22no—
266C2×C32—21no—
267C2×C32—41no—
268C2×C32—42no—
269C2×C32—42no—
270C2×C32—42no—
271C2×C32—44no—
272C2×C32—44no—
273C2×C32—84no—
274C2×C32—84no—
275C2×C32D3×C22⋊C412yes—
276C2×C32C2×#21711yescharacteristic
277C3⋊D3—22no—
278C3×D3—64no—
279C3×D3—122no—
280#41—182no—
281#41—182no—
282C2×Q8—91no—
283C2×D4—92no—
284C2×D4#111311yescharacteristic
285C22×C4—91no—
286C22×C4—92no—
287M16—94no—
288M16—542no—
289C2×C8—542no—
290C22×C3—24no—
291C22×C3—24no—
292C22×C3—22no—
293C22×C3—21no—
294C22×C3—22no—
295C22×C3—44no—
296C22×C3—44no—
297C22×C3—44no—
298C22×C3—42no—
299C22×C3—44no—
300C22×C3—41no—
301C22×C3—84no—
302C22×C3—84no—
303C22×C3—92no—
304C22×C3—361no—
305C22×C3C2×D3×Dic312yes—
306C22×C3C2×#62411yescharacteristic
307C2×D3—34no—
308C2×D3—62no—
309C2×D3—122no—
310C2×D3—122no—
311C3×C4—24no—
312C3×C4—22no—
313C3×C4—42no—
314C3×C4—91no—
315C3×C4—91no—
316C3×C4—182no—
317C3×C4D3×#6714yes—
318C3×C4#381112yes—
319Dic3—32no—
320Dic3—32no—
321Dic3—61no—
322Dic3—61no—
323Dic3—64no—
324Dic3—64no—
325C32—22no—
326C32—21no—
327C32—41no—
328C32#474512yescomplemented
329C32#523711yescharacteristic, complemented
330C23—91no—
331C23#629412yes—
332Q8—182no—
333D4—22no—
334D4—184no—
335C2×C4—91no—
336C2×C4—92no—
337C2×C4—92no—
338C2×C4—92no—
339C2×C4—182no—
340C2×C4—182no—
341C2×C4C2×#111311yescharacteristic
342C8—184no—
343C2×C3—22no—
344C2×C3—22no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—22no—
348C2×C3—42no—
349C2×C3—42no—
350C2×C3—44no—
351C2×C3—44no—
352C2×C3—42no—
353C2×C3—41no—
354C2×C3—84no—
355C2×C3—84no—
356C2×C3—182no—
357C2×C3D3×#21712yes—
358C2×C3#6397411yescharacteristic
359D3—64no—
360D3—122no—
361C22—22no—
362C22—42no—
363C22—92no—
364C22—361no—
365C22C2×#629411yescharacteristic
366C4—91no—
367C4—91no—
368C4—182no—
369C4#8904912yes—
370C3—21no—
371C3—22no—
372C3—41no—
373C3#1059150112yescomplemented
374C3#1059150411yescharacteristic, complemented
375C2—21no—
376C2—42no—
377C2—182no—
378C2#1172372311yescharacteristic
subgroupG/H
centerC2#11723723
commutatorC22×C33C22×C4
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3830C2
cosocleC2×C33×C4C23
hypercenterC2×D4#1113
hypocenterC33#413
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11723656C2semidirect
3. C3×#63660C2semidirect
4. #11722074C2semidirect
22. #63660D3semidirect
215. C33#413semidirect
328. C32#4745semidirect
329. C32#5237semidirect
373. C3#10591501semidirect
374. C3#10591504semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#38302
#117236562
C3×#636602
#117220742
#105915043
#105915013